Evaluate Gamma Integral: j,k Positive Constants

It appears in the integral for the autocorrelation of the Riemann zeta function, and is related to the Hurwitz zeta function.In summary, we discussed the idea of evaluating the integral \lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt, which cannot be done as it is not convergent for T approaching infinity. We also mentioned the use of a Fourier transform, but determined it would not be helpful in this case. Additionally, we touched on the properties of the gamma function ratio, specifically its appearance in the autocorrelation of the
  • #1
mmzaj
107
0
greetings . any ideas on how to evaluate this integral

[tex] \lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt[/tex]

[itex] j, k [/itex] are positive constants .
 
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  • #2
mmzaj said:
greetings . any ideas on how to evaluate this integral

[tex] \lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt[/tex]

[itex] j, k [/itex] are positive constants .

Hi !
this integral cannot be evaluated for T-->infinity because it is not convergent.
 

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  • #3
The gamma function ratio is a polynomial in t (degree = j, if j is an integer).
 
  • #4
mmzaj said:
greetings . any ideas on how to evaluate this integral

[tex] \lim_{T\rightarrow \infty}\frac{1}{2T}\int_{-T}^{T}\frac{\Gamma(3+it)}{\Gamma(3+it-j)}e^{ikt}dt[/tex]

[itex] j, k [/itex] are positive constants .

Not sure, but would a Fourier transform help here?
 
  • #5
KarmonEuloid said:
Not sure, but would a Fourier transform help here?
I don't think so. Since the integral is not convergent for T tending to infinity, the Fourier transform is of no help to find a limit which doesn't exist anyways.

mathman said:
The gamma function ratio is a polynomial in t (degree = j, if j is an integer).
This is a so-called Pochhammer polynomial.
 

1. What is the Gamma Integral?

The Gamma Integral is a mathematical function that is used to solve problems involving areas under curves. It is defined as the integral of a function over a range of values from 0 to infinity.

2. How is the Gamma Integral evaluated?

The Gamma Integral can be evaluated using various methods, such as numerical integration or the use of special functions. The specific method used will depend on the complexity of the function being integrated and the desired level of accuracy.

3. What are j and k in the context of the Gamma Integral?

In the context of the Gamma Integral, j and k are positive constants that are used as parameters in the function. They can take on any positive value and are typically chosen based on the problem being solved.

4. What is the significance of positive constants in the Gamma Integral?

The use of positive constants in the Gamma Integral allows for a more general and flexible solution to problems involving areas under curves. It allows for a wider range of functions to be integrated and provides more control over the shape of the curve.

5. What are some real-world applications of the Gamma Integral?

The Gamma Integral has numerous applications in fields such as physics, engineering, and economics. It can be used to solve problems involving population growth, radioactive decay, and financial modeling, to name a few.

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